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## Kinetic theory

Lesson for 16-19

Students will have been introduced to a simple kinetic model of matter in both chemistry and physics. They will be able to use it, for example, to explain how pressure of a gas arises, and perhaps understand the nature of temperature. Here, they can learn more about how the microscopic behaviour of particles gives rise to macroscopic behaviour; they can also see how this gives rise to a mathematical model.

## Episode 600: Preparation for kinetic theory topic

Teaching Guidance for 16-19

- Level Advanced

Here, students can learn more about how the microscopic behaviour of particles gives rise to macroscopic behaviour; they can also see how this gives rise to a mathematical model.

#### Main aims of this topic

Students will:

- appreciate some of the experimental evidence for atoms and their motion
- know how to apply the gas laws
- understand the thermodynamic temperature scale
- know what an ideal gas is, understand how its properties link to the gas laws and the absolute temperature scale, and be able to link absolute temperature with average energy per molecule
- know that the energy in a gas is not uniformly distributed among its particles and be able to make calculations using the Boltzmann constant

#### Prior knowledge

Students should already know sufficient kinematics and dynamics (including momentum) to be able to derive the ideal gas equation if this section is to be proved formally in your specification.

#### Where this leads

These episodes end with a consideration of ways in which energy can be supplied to a substance (by heating or doing work). This can lead on to a consideration of the laws of thermodynamics.

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### Brownian motion and ideal gases

## Episode 601: Brownian motion and ideal gases

Lesson for 16-19

- Activity time 140 minutes
- Level Advanced

This episode looks at Brownian motion as evidence for the particulate nature of matter, and the macroscopic gas laws.

Lesson Summary

- Demonstration and discussion: Brownian motion and what this tells us about air (and other gases) (30 minutes)
- Demonstration and student experiments: The gas laws (60 minutes)
- Discussion: Boyle’s law – a particle explanation (20 minutes)
- Discussion: Extrapolating to absolute zero (10 minutes)
- Student activity: A computer model of Boyle’s law (20 minutes)

#### Discussion and demonstrations: Brownian motion and what this tells us about air (and other gases)

A reasonable place to start is by reviewing the evidence that matter is made up of particles (molecules and atoms). (It is probably best to refer to particles

in general, and to think of them as spherical; point out that, by this, we mean either atoms or molecules.) Students will have met these ideas at a lower level in chemistry as well as physics. Evidence includes the combination laws of gases, and Brownian motion, which can be demonstrated in the classroom.

According to your students’ previous experience, you may wish to demonstrate Brownian motion, the expansion of bromine into a vacuum, and a measurement of the density of air.

Episode 601-1: Brownian motion in a smoke cell (Word, 58 KB)

Episode 601-2: Diffusion of bromine (Word, 72 KB)

Episode 601-3: The density of air (Word, 72 KB)

Brownian motion is evidence not just for the existence of atoms or molecules but also for their movement, which is random. This random motion can be modelled mathematically and leads to a test for the size of the atoms from measured diffusion rates – this was one of Einstein’s great papers of 1905.

Question: What is the mass of air in the room? Answering this will require the estimation of the volume of the room, and the use of the density of air. Students are often surprised by the result: perhaps 100 kg, more than the mass of a typical student. If all of the air in the room condensed to form a liquid, it would make a layer perhaps 5 mm deep on the floor.

Now you can introduce a gas as a simple system. In crystalline solids, all the atoms are nicely ordered in an array, making calculations possible. In a gas, the motion is random and again this simplifies calculations as the laws of statistics can be applied. (Disordered solids and liquids are more difficult to treat mathematically, because they are neither well-ordered nor completely disordered.)

#### Demonstration and student experiments: The gas laws

Now move on to the gas laws. Strictly speaking it is only necessary to look at Boyle’s law (*P**V* = constant at constant temperature).

You may well have an apparatus specifically designed to show this such as an oil-filled column attached to a pump and pressure gauge. You pump air in to pressurise the oil, which compresses an air space at the top. A series of about 10 *P* and *V* readings usually gives a good fit to a straight line when *P* is plotted against 1*V*.

Episode 601-4: Changes in volume, changes in pressure (Word, 58 KB)

The other laws follow from this and the definition of *thermodynamic* temperature. However, at this stage it is likely that the only idea of temperature students are familiar with is that which is measured by a thermometer

. (This is not a totally silly definition as it relies on the fact that when two bodies of different material, temperature and size are in contact, their temperatures equalise. This is usually referred to as the Zeroth Law of Thermodynamics.)

So it is worth pressing ahead with demonstrations or class experiments of Charles’ law (*V* proportional to *T*) and/or the pressure law (*P* proportional to *T*).

Episode 601-5: Changes in temperature, changes in pressure (Word, 96 KB)

Episode 601-6: Changes in temperature, changes in volume (Word, 112 KB)

#### Discussion: Boyle’s law – a particle explanation

You may need to discuss how Boyle’s law can be explained in terms of particles.

Episode 601-7: Boyle’s law, density and number of molecules (Word, 65 KB)

#### Discussion: Extrapolating to absolute zero

Both Charles’ law and the pressure law lead to the concept of an absolute zero of temperature. Note that Absolute Zero (0 K) is the temperature at which the energy of the particles of a material has its minimum value; this is not zero, as the particles have so-called zero point energy due to quantum effects, which cannot be removed. They still vibrate.

Remember real gases turn into liquids and solids before absolute zero is reached. However extrapolating back often gives reasonable values.

Episode 601-8: Changing pressure and volume by changing temperature (Word, 53 KB)

Episode 602: Ideal gases and absolute zero (Word, 56 KB)

#### Student activity: A computer model of Boyle’s law

If suitable apparatus for demonstrating Boyle’s law is not available it can be simulated using computer packages or applets but this is a poor relation to the real experiment and should only be used to illustrate the effect, not to demonstrate it.

See a model on the

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### Up next

### Ideal gases and absolute zero

## Episode 602: Ideal gases and absolute zero

Lesson for 16-19

- Activity time 75 minutes
- Level Advanced

This episode establishes the ideal gas law, and how to use it. NB many texts use the terms ideal gas and perfect gas as meaning the same thing. Strictly speaking an ideal gas is one that obeys Boyle’s law with complete precision. A perfect gas is a real gas under conditions that Boyle’s law is a valid enough description of its behaviour.

Lesson Summary

- Discussion: The ideal gas law, moles and the Kelvin scale of temperature (30 minutes)
- Worked examples: Using the equation; temperature conversions (15 minutes)
- Student questions: Practice questions (30 minutes)

#### Discussion: The ideal gas law, moles and the Kelvin scale of temperature

Both Charles’ law and the pressure law lead to an extrapolation back to zero volume or pressure which would imply the temperature scale can go no further. For all gases, that zero point (absolute zero) is (roughly) the same and although clearly the gas would no longer be a gas there, this is an important implication.

The separate laws can be combined into the ideal gas law,
*P**V* = N*R**T*. Make sure that your students understand the different symbols in this equation: N is the number of moles of gas and *R*
is the molar gas constant, i.e. the gas constant for one mole of substance, with the value 8.3 J^{K-1}.

It may be necessary to review these ideas. Many students will have learned that 1 mole of gas occupies 22.4 litre (at STP) or 24 litre (at RTP). It will be necessary to break them out of the habit of using this shortcut in order to apply the ideal gas law correctly. 1 mole is simply a standard number of atoms, Avogadro’s number, 6.023 × 10^{23}. The volume of one mole follows from the ideal gas law and the molar gas constant. They should also recall how to calculate the number of moles of a substance from

n = massrelative molecular mass.

The ideal gas law is approximately true for most gases, and as its name implies is exactly true for an ideal gas, an imaginary gas which obeys Boyle’s law perfectly.

Such a gas can be used to define the thermodynamic temperature scale with its zero where *P* and *V* drop to zero. In practice we know that there is a small amount of energy at absolute zero, the so-called zero-point motion of quantum mechanics. So it is best to define absolute zero as the point of lowest energy not zero energy.

To be strictly accurate, the Kelvin and Celsius scales coincide at the triple point of water, which is 0.01 ° C or 273.16 K, and so the conversion between the two scales is:

temperature in K = temperature in °C + 273.15.

The conventional symbols are:

*T*for thermodynamic (absolute) temperature, SI unit: kelvin, symbol K-
*q*for temperature in °C.

#### Worked examples: Using the ideal gas law; temperature conversions

It is necessary to emphasise that the proportionality laws only apply with absolute temperature, so make sure that your students know when to work in K and how to convert between ° C and K.

Show a worked example or two of this.

Also, work through an example or two using the ideal gas law.

Episode 602-1: Ideal gases (Word, 20 KB)

#### Student questions: Practice questions

Practice questions on gas laws, moles, absolute zero.

Episode 602-2: Using the ideal gas relationships (Word, 31 KB)

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### Kinetic model of an ideal gas

## Episode 603: Kinetic model of an ideal gas

Lesson for 16-19

- Activity time 100 minutes
- Level Advanced

This episode relates the gas laws to the behaviour of the particles of a gas.

Lesson Summary

- Discussion and demonstration: explaining pressure in terms of particles (15 minutes)
- Discussion: deriving an equation for the pressure of a gas (30 minutes)
- Discussion: the link between energy and pressure (20 minutes)
- Worked example and student questions: Calculating molecular speeds (20 minutes)
- Discussion: internal energy of a gas (15 minutes)

#### Discussion and demonstrations: Explaining pressure in terms of particles

In the previous episode, we looked at the macroscopic behaviour of a gas, in terms of its temperature, pressure and volume. Now we can go on to relate this behaviour to the underlying microscopic behaviour of the particles of which the gas is made.

Remind your students of the description of pressure arising from bombardment by particles of the walls of the container. Ask how the pressure will change if:

- the particles move faster (i.e. higher
*T*)? - the particles have greater mass
*m*? - there are more particles?

(All of these will result in greater pressure. You may wish to point out at this stage that increasing v has two effects: a greater force on impact, and more frequent impacts.)

You can usefully demonstrate the heating effect when the air in a bicycle pump is compressed.

Episode 603-1: Warming up a gas by speeding up its particles (Word, 46 KB)

#### Discussion: Deriving an equation for the pressure of a gas

pV=13Nm{*c*^{ 2}}

The discussion above will prepare your students for the derivation and use of the equation for the pressure of an ideal gas, where N is the total number of molecules in the volume *V*, and the bar indicates an average. Check whether your specification requires the derivation. It is covered in most of the major texts. Even if the derivation is not required, you will have to explain the terms used, and show that the equation is plausible. Point out that the quantities on the left are macroscopic, while those on the right are microscopic.

Stress the underlying assumptions of the model. The gas:

- has zero volume at zero temperature so the volume of the actual molecules is negligible
- has zero pressure at zero temperature so heating is the only way to get the molecules moving
- have atoms or molecules which behave as elastic spheres with no long-range intermolecular forces

There are a number of points to look out for in the derivation. Students will need convincing about the idea of averaging the *square* of a velocity; the average velocity is zero, because particles with opposite velocities cancel out. The change in momentum is 2*m**v* when an atom rebounds. The square of the velocity arises because *v* increases both the momentum change and the frequency of impact. We can average a series of impulses into a smooth net force to give an impulse Ft. All of this will rely on having covered momentum and impulse thoroughly.

If necessary try a demonstration to help convince the students. Balls strike a force sensor at increasing frequencies and with increasing velocity. The force sensor records the pressure which results.

Episode 603-2: One collision: many collisions (Word, 97 KB)

#### Discussion: The link between energy and pressure

You can now compare the expression:

pV=13Nm{*c*^{ 2}}

With the ideal gas law to show the equivalence between temperature and the average energy per molecule (Given the elastic sphere assumption, this energy is only stored kinetically). This idea follows directly from a statistical analysis of elastic collisions but the maths involved is too advanced for the post-16 level. However the idea may be demonstrated by computer models that have programmed into them only the laws of dynamics for collisions, and nothing about thermodynamics. Such programs demonstrate the plausibility of the idea and are a powerful visual stimulus.

Episode 603-3: Kinetic theory applets (Word, 24 KB)

If your specification requires it, make that link formal with:

12*m**v*^{ 2}=32*k**T*

Where *k* is Boltzmann’s Constant, and
*k* = *R**N*_{A}.

#### Worked example and student questions: Calculating molecular speeds

This equation allows us to calculate mean molecular speeds, which should be done as a worked example. Calculate the rms speed for oxygen molecules at room temperature; ask the class to repeat the calculation for nitrogen, or for other molecules in air. They will find that the lighter the molecule, the faster its particles move.

Compare these speeds with the speed of sound in air (~330 m s^{-1}), and with the Earth’s escape velocity (11 km s^{-1}). You could also draw a comparison with familiar diffusion rates (e.g. speed with which a stink bomb is detected), leading to the idea of collisions and mean free path, which need not be entered into too deeply. The following question would be excellent to work through, leaving students to do some parts, and helping them through trickier concepts such as ratios.

Episode 603-4: Speed of sound and speed of molecules (Word, 58 KB)

#### Discussion: Internal energy of a gas

- Energy flowing into a liquid initially raising the temperature, then boiling it, then heating it as a gas. All stages increase internal energy
- Joule’s paddle wheel experiment where instead of heating increasing the internal energy, mechanical working is converted directly to internal energy. The temperature is raised and the equivalence of heating and doing mechanical work as two means of transferring energy is demonstrated. This is leading towards the first law of thermodynamics
- Cooling something down by evaporation. Compare when it is on a surface that then supplies further energy to keep the temperature constant (people sweat to cool down) and evaporation when there is no such reheating, so that the liquid cools (puddles evaporate away totally). A drop of isopropyl alcohol on the back of the hand demonstrates the cooling effect clearly – it feels cold as it evaporates

You are now in a position to talk about *internal energy* as the total energy stored kinetically and chemically in the molecules. Energy that is stored chemically in the bonds must be considered as you need to transfer energy to melt a solid or boil a liquid; this does not go into the energy stored kinetically by the molecules as the temperature stays constant. Another key idea here is that the energy calculated from the temperature using
32*k**T* = 12 *m**v*^{ 2}
is an average and, therefore, there must be a distribution: some particles will have greater than this and some less. This leads to explanations of everything from evaporation to chemical reaction rates.

All of the above are quite subtle, sophisticated ideas that require the students to absorb definitions of closely related concepts. As such it cannot be rushed, and discussion should be spaced out with examples.